Q1. State the definition of a function.

Background

Topic: Functions and Their Definitions

This question tests your understanding of what a function is in mathematics, which is foundational for all of precalculus.

Key Terms:

• Function: A relation that assigns exactly one output for each input from a given set.

• Domain: The set of all possible input values (usually x-values).

• Range: The set of all possible output values (usually y-values).

Step-by-Step Guidance

1. Recall that a function relates each element of the domain to exactly one element of the range.

2. Think about how you would describe this relationship in your own words, emphasizing the "one output for each input" rule.

3. Consider using set notation or mapping notation if you are comfortable with it, but a clear sentence is sufficient.

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Q2. Given that , what is the domain of ?

Background

Topic: Domain of Rational Functions

This question tests your ability to find the domain of a rational function, which involves identifying values that make the denominator zero.

Key Terms and Formulas:

• Rational Function: A function of the form where .

• Domain: All real numbers except those that make the denominator zero.

Step-by-Step Guidance

1. Identify the denominator of the function: .

2. Set the denominator equal to zero: .

3. Solve for to find the value(s) that are not allowed in the domain.

4. Express the domain in interval notation, excluding the value(s) found in the previous step.

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Q3. Use the TI-83 or TI-84 calculator to solve the equation . All solutions lie between -10 and 10. Approximate the solutions to two decimal places.

Background

Topic: Solving Polynomial Equations Numerically

This question tests your ability to use a graphing calculator to find approximate solutions (roots) of a higher-degree polynomial equation.

Key Terms and Concepts:

• Root (Zero): A value of where .

• Graphing Calculator: Use the 'zero' or 'root' function to find where the graph crosses the x-axis.

Step-by-Step Guidance

1. Enter the equation into your calculator's Y= menu.

2. Set the window to values between -10 and 10 to ensure you see all possible roots in the given interval.

3. Use the 'zero' or 'root' function on your calculator to find the x-values where the graph crosses the x-axis.

4. Record each solution, rounding to two decimal places as required.

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Q4. Given the points and :

Background

Topic: Linear Equations and Distance Formula

This question tests your ability to find the equation of a line given two points and to calculate the distance between those points.

Key Terms and Formulas:

• Slope-Intercept Form:

• Slope Formula:

• Distance Formula:

Step-by-Step Guidance

1. Find the slope using the two points: and .

2. Use the slope and one point to write the equation in point-slope form, then convert to slope-intercept form.

3. For the distance, substitute the coordinates into the distance formula and simplify under the square root.

4. Leave the distance in exact form (do not approximate).

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Q5. Given and , find each of the following:

Background

Topic: Function Evaluation and Operations

This question tests your ability to evaluate functions, add and subtract functions, and find intercepts.

Key Terms and Formulas:

• Function Evaluation: Substitute the given value for in the function.

• Sum of Functions:

• Difference of Functions:

• y-intercept: The point where .

Step-by-Step Guidance

1. For , substitute into and simplify.

2. For , add and together and combine like terms.

3. For , substitute into both and , then subtract from .

4. For the y-intercept of , substitute into and write the coordinates as .

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Q6. Solve each inequality or equation. Exact values only. Reduce fractions, and provide your answer using interval notation where appropriate.

Background

Topic: Absolute Value Equations and Linear Inequalities

This question tests your ability to solve equations involving absolute values and inequalities, and to express solutions in interval notation.

Key Terms and Formulas:

• Absolute Value Equation: leads to two equations: and .

• Linear Inequality: Solve as you would a linear equation, but reverse the inequality if you multiply or divide by a negative.

Step-by-Step Guidance

1. For , set up two equations: and .

2. Solve each equation for .

3. For , distribute and combine like terms.

4. Isolate and express the solution in interval notation.

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Q7. Solve the system of equations:

Background

Topic: Systems of Linear Equations

This question tests your ability to solve a system of two linear equations using substitution or elimination.

Key Terms and Formulas:

• System of Equations: Two or more equations with the same variables.

• Substitution Method: Solve one equation for one variable and substitute into the other.

• Elimination Method: Add or subtract equations to eliminate a variable.

Step-by-Step Guidance

1. Solve the second equation for in terms of .

2. Substitute this expression for into the first equation.

3. Solve for .

4. Substitute the value of back into one of the original equations to find .

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Q8. Use the graph of the given function to answer the following questions. The turning points and x-intercepts have been labeled.

Background

Topic: Analyzing Graphs of Functions

This question tests your ability to interpret key features of a function's graph, including degree, extrema, intercepts, and intervals of increase/decrease.

Key Terms and Concepts:

• Degree: The highest power of in the function.

• Local Maximum/Minimum: Highest/lowest point in a local region of the graph.

• Increasing/Decreasing Intervals: Where the graph rises or falls as increases.

• : Where the graph is below the x-axis.

Step-by-Step Guidance

1. Look at the end behavior and number of turning points to determine if the degree is even or odd.

2. Identify the highest labeled point (local maximum) and lowest labeled point (local minimum) on the graph.

3. Find the intervals where the graph is rising (increasing) and where it is falling (decreasing).

4. Determine the intervals where the graph is below the x-axis ().

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Q9. Match each equation to its graph.

Background

Topic: Parent Functions and Their Graphs

This question tests your ability to recognize the graphs of basic functions such as linear, quadratic, cubic, exponential, and reciprocal functions.

Key Terms and Concepts:

• Parent Function: The simplest form of a function type.

• Common Parent Functions: , , , ,

Step-by-Step Guidance

1. Recall the general shape of each parent function listed.

2. Match each equation to the graph that best fits its shape (e.g., parabola for , S-curve for , etc.).

3. Be careful with the reciprocal and exponential functions, as their graphs are distinct.

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Q10. a) What transformations have been applied to the parent function to arrive at ?

Background

Topic: Transformations of Functions

This question tests your understanding of how to describe transformations such as shifts, stretches, and translations applied to a parent function.

Key Terms and Concepts:

• Vertical Stretch: Multiplying the function by a constant greater than 1.

• Horizontal Shift: Adding or subtracting inside the function argument.

• Vertical Shift: Adding or subtracting outside the function.

Step-by-Step Guidance

1. Identify the coefficient in front of the squared term to determine if there is a vertical stretch or compression.

2. Look at the value subtracted from inside the parentheses to determine the horizontal shift.

3. Look at the value added outside the squared term to determine the vertical shift.

4. Describe each transformation in order.

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